Why is the eak needed exactly? Or they just trying remind us that eek can be used for " a game". I am not clear about the rules/ guidelines for when to use or not use eeak.
Loosely seems to be that 'the' ('the game') is implicit, unless एक makes it 'a'. But there's some variation because what's definite or not varies across languages, colloquialisms, and contexts.
Why not " me ek khel khelthi hun "
That would apparently be "I play a game"
खेल खेल ??
verb and noun the same? is that right?
Yes, खेलना is the verb's infinitive, and खेल is the noun (and verb stem). Similarly in English: 'who is playing [verb] the lead role in that play [noun]', and with many other words.